Two states that hired the same lead contractor to build their online marketplaces for health insurance under President Barack Obama’s Affordable Care Act have experienced so many problems and delays that they are threatening to withhold millions of dollars in payments to the company and even seek refunds.

Massachusetts health officials said they will meet in early January to decide whether to suspend payments to the company, CGI Federal, which has also been criticized — and grilled during a congressional hearing — for its central role in building the problem-plagued federal health insurance exchange. The state may also seek a refund from the firm, which is an American subsidiary of the Canadian-based CGI Group.

Already, Vermont has refused to pay $5.1 million of its contract with the company and is trying to get hundreds of thousands of dollars refunded. The two states’ problems with CGI were reported by the Boston Globe.

Under the Affordable Care Act, eligible Americans who do not have health insurance through their workplace or another source are required to sign up for coverage through a health care exchange that will provide a menu of options and information on subsidies. Open enrollment runs through March; those who have not signed up by then could face penalties.

Fourteen states, including Massachusetts and Vermont, created their own online marketplaces, and the rest depend on the exchange operated by the federal government.

While the federal site had a rocky rollout on Oct. 1, several of the state exchanges also stumbled. In Hawaii, the site created by CGI performed so poorly that the head of the exchange stepped down this month. Other exchanges — including Kentucky’s and California’s, on which CGI worked as a subcontractor — have performed relatively well.

“Right now, our current focus is ensuring there are no gaps or delays in coverage for our members,” said Jason Lefferts, a spokesman for Massachusetts Health Connector, the state’s insurance exchange. “We will be coming back to the board in January to discuss further accountability for the IT vendor and also talk about the path forward.”

Massachusetts has already paid CGI $11 million under a $69 million contract.

Lefferts said that the exchange continued to perform slowly, and that people had “recurring problems” creating accounts and logging in. “We still don’t have the automated ability to process an application and determine someone’s eligibility,” he said. “We also don’t have the ability to send enrollment information directly to the insurance carrier.”

Lorne Gorber, senior vice president for global communications and investor relations at the CGI Group, said: “The goal of the clients and the goal of ourselves is the same: That’s to get everything right and not be wasting a lot of time and effort in the clauses of the contracts when we’re doing that.”

Noting that CGI is involved in nine state exchanges, he said, “It’s expected that some of them are not the best performing ones.”

“It’s important to note that we’re not at the bottom of the worst performing ones,” he added, “and we’re at the top of the list of the best performing ones.”

The exchange’s performance has been especially nettlesome in Massachusetts, which had its own online health insurance system before the Affordable Care Act. Because the federal law required so many extra features, the state decided to build a new website — which is now malfunctioning.

In Vermont, Gov. Peter Shumlin has frequently expressed frustration with his state’s insurance exchange, complaining of repeated delays and glitches since its unveiling.

“I won’t tolerate a situation where Vermonters go into the holiday season worried and confused by their health care options come January 1st,” the governor said in late October, when problems with the state website forced him to delay some enrollment deadlines.

Continued glitches, though, have prompted Vermont to hold back a $5.1 million payment and threaten to withhold more if the problems are not fixed. So far, the state has paid only $19 million of its $83 million contract with CGI.

Mark Larson, Vermont’s commissioner of health access, sent CGI two letters last month that spell out the various ways the company had failed to deliver on its contract, including missing critical deadlines and charging for services never performed.

“Most importantly,” Larson wrote, “CGI has yet to provide the state of Vermont with a fully functioning website. Vermont residents still cannot buy their health plans online, as promised.”